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Probability Question

Here's the question:

In a city there are 3 motorcycle sellers. Each seller make it's own motorcycle and sell it to people. In a day about 1000 motorcycles are sold in the city. The probability that a seller's bike would be sold is given by \(P_n\) where n is the particular seller. Given \(P_1 = 2/8\), \(P_2 = 4/8\), \(P_3 = 6/8\), find the expected number of motorcycles sold by each seller.

I made this question. I am not sure if the question is technically correct or not. I further doubt if \(P_1 + P_2 + P_3 = 1\) is a necessary condition or not.

Help me figuring it out.

Note by Lokesh Sharma
3 years ago

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It is consistent, but incomplete. \(P_1 + P_2 + P_3 \neq 1\) is all fine; what we know is that "among the bikes that the first seller sells, it's expected that \(P_1\) of them are actually sold", and so on. We're missing the number of bikes that each seller plans to sell. However, assuming the numbers are identical, we can solve it:

Suppose each seller makes \(x\) motorcycles. Then seller \(i\) is expected to sell \(P_ix\) motorcycles. In total, there are \(1000\) motorcycles sold, so \(P_1x + P_2x + P_3x = \frac{3x}{2} = 1000\), giving \(x = \frac{2000}{3}\). Thus the expected number of motorcycles sold by the first seller will be \(P_1x = \frac{500}{3}\), and similarly with other sellers. Ivan Koswara · 2 years, 12 months ago

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@Ivan Koswara Oh thank you very much! It was easier then I expected; my teacher almost took the entire lecture and couldn't figure it out. Lokesh Sharma · 2 years, 12 months ago

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